Let curves be defined as parametric equations of form $R=(y(t),x(t))$ or the set of solutions for the equations $F(x,y)=0$ where $y(t)$, $x(t)$, $F(x,y)$ are a combination of any algebraic functions and transcendental functions.
Can we obtain the whole set of differentiable planar curves from all of the combinations of $y(t)$, $x(t)$ and $F(x,y)$?
In the same manner as the Complex numbers are enough (we don't need any set bigger than $\Bbb C$) to represent all solutions of any polynomial equation even if we don't have a general formula to find all the solutions for polynomials of degree bigger than 5, is it known whether all the combinations of algebraic and transcendental functions for $y(t)$, $x(t)$ and $F(x,y)$ are enough to cover ALL differentiable planar curves even if there is no way to generate them? Or do we need even more functions such as hypertranscendentals and above if there are any?
Some examples of what I mean to generate all curves:
We can definitely define the unit circle: $x^2 + y^2 - 1 = 0$
But can we obtain the circle with a small bump?
Or 2 small bumps?
Or all sizes of bumps?
Can we construct everything with careful alterations of existing definitions of $y(t)$, $x(t)$ and $F(x,y)$?
Notes:
I used the arc length integral formula to obtain the bumpy circles graphs. Is that formula considered a hypertranscendental function? If yes then I guess that answers half of my question -> algebraic and transcendental functions are not enough to cover all the planar curves. But is it enough to add just hypertranscendental functions in the mix?
I won't post the definitions for the curves in the images as they are way too complicated with hundreds of terms.
Extra questions:
Can we achieve the whole set of planar curves WITHOUT using piecewise notation for the definitions? Having piecewise definitions in the parametric equation case for different input intervals kind off defeats the purpose of this whole question. One could argue that if piecewise definitions are allowed then all infinity of planar curves can be achieved by those definitions that have an infinite number of piecewise branches.
What other function types are there besides the already enumerated ones (algebraic, transcendental, hypertranscendental) and how can those contribute to "filling" the set of planar curves?
Are there ways to define curves other than parametric or implicit equations?
The same as above but for curves in 3d space.
The same as above but for surfaces in 3d space.
This works for algebraic functions, but not for transcendentals $F(x,y)=0$. Every polynomial $F(x,y)=0$ has a parametrization (that is, $F(f,f')=0$) in terms of Elliptic functions (for some Elliptic function $f$) and every Elliptic function can be written as a rational function of the Weierstrass p-function $\wp(z)$ and its derivative. In fact, for a polynomial $F(x,y)$ the parametrization $(f,f')$ of its zero locus can be written explicitly using the fact that $$f=\frac{P_1(\wp(z))+P_2(\wp(z))\wp'(z)}{P_3(\wp(z))},$$ for polynomials $P_1,P_2,P_3$ to be determined.
As for
No... yes? I mean you have to define the space in which you are working and put a condition/restrict that space by a condition or set of conditions, whether implicitly or explicitly.
I'm sure you could get clever with the theorem that $F(x,y)=0$ is parametrized by $(x,y)=(f,f')$ and figure out a way to get $F(f,f_u,f_{uu})=0$ going.
Ibid, only $F(f,f_u,f_v)=0$.
$$\wp'^2=4\wp^3-g_2\wp-g_3$$ and $$\wp=\frac{1}{z^2}+ \sum_{\omega\in\Lambda}'\frac{1}{(z-\omega)^2}-\frac{1}{\omega^2}$$ and $$g_2:=60\sum_{\omega\in\Lambda}'\frac{1}{\omega^4}\\ g_3:=140\sum_{\omega\in\Lambda}'\frac{1}{\omega^6}.$$ Choose a doubly periodic lattice in the complex plane $\Lambda.$